Obsidian Vault
NOTES-MARKDOWN
Linear Algebra
Basis
Basis
HTML
Determinant
HTML
Determinant
Dot product
Dot product
HTML
Eigenvectors and Eigenvalues
Eigenvectors and Eigenvalues
HTML
Gaussian Elimination
Gaussian Elimination
HTML
Hyperplanes
HTML
Hyperplanes
Inverse of a Matrix
Inverse of a Matrix
HTML
Linear Combination
HTML
Linear Combination
Linear dependence
Linear dependence
HTML
Matrix (ML)
Matrix (ML)
HTML
Matrix operations
Matrix operations
HTML
Norm
HTML
Norm
Null space
HTML
Null space
Quadratic form
Quadratic form
HTML
Rank
Rank
HTML
Rayleigh quotient
Rayleigh quotient
HTML
Spanning set
Spanning set
HTML
Spectral decomposition
HTML
Spectral decomposition
SVD
HTML
SVD
Symmetric matrix
HTML
Symmetric matrix
Systems of Linear Equations
HTML
Systems of Linear Equations
Transformations
Transformations
HTML
Vector spaces
HTML
Vector spaces
Vectors
HTML
Vectors

Inverse of a Matrix

Let be a square matrix, if there exists a square matrix such that...

... then we say that is invertible and is the inverse of ().

Note

A matrix that doesn't have an inverse is called singular or non-invertible.

Finding the inverse of a Matrix

The Gauss-Jordan elimination method can be used to find the inverse of a matrix. Here's a step-by-step process:

  1. Augment the Matrix with the Identity Matrix.

  2. Apply Row Operations to Reach RREF: Use the Gauss-Jordan elimination method to transform the given matrix into the reduced row echelon form (RREF). The goal is to transform the left side of the augmented matrix into the identity matrix.

Hint

Use elementary row operations, including swapping rows, multiplying a row by a nonzero scalar, and adding/subtracting a multiple of one row from another row.

  1. Extract the Inverse: Once the left side of the augmented matrix is the identity matrix, the right side will be the inverse of the original matrix.
Warning

Note that this method only works for square matrices that are invertible.
If the given matrix is not invertible (i.e., it has no inverse), the process will reveal this fact by leading to a row of zeros or inconsistencies in the augmented matrix.

Example

Suppose we want to find the inverse of the following matrix:

  1. Augment with the Identity Matrix:

  2. Row Operations:

    • Multiply the first row by :
    • Subtract the first row from the second row:
    • Multiply the second row by 2:
    • Subtract times the second row from the first row:

  3. Extract the Inverse:
    The inverse of is:

Properties of Inverse Matrices

Existence of the Inverse:
The inverse of a matrix exists if and only if is square and has a non-zero determinant.

Hint

A non-zero determinant indicates that the matrix represents a transformation that doesn't "collapse" space; hence, it can be reversed.

Transposition of an Inverse:
If is invertible, then .

Hint

Transposing a matrix and finding its inverse are reversible operations, so the order in which you perform them doesn't matter.

Inverse of a Scalar Multiple:
If is invertible and is a nonzero scalar, then .

Hint

Scaling a matrix by a factor scales its inverse by , as the scaling must be "undone" in the inverse transformation.

Exercise

Inverse of a product

Let A and B be invertible matrices, then AB is also invertible and

Exercise

System of Equations with Unique solution

Intuitively, you can solve the an equation of the form with .

You can use the same method for solving system of linear equations of the form...

Where A is an invertible matrix.

Hint

Exercise (using Gauss-Jordan elimination)

Table Of Contents
Inverse of a Matrix
Finding the inverse of a Matrix
Properties of Inverse Matrices
Inverse of a product
System of Equations with Unique solution